► In 1949, Wever observed that the degree d of an invariant Lie polynomial must be a multiple of the number q of generators of the…
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▼ In 1949, Wever observed that the degree d of an invariant Lie polynomial must be a multiple of the number q of generators of the free Lie algebra. He also found that there are no invariant Lie polynomials in the following cases: q = 2, d = 4; q = 3, d = 6; d = q ≥ 3. Wever gave a formula for the number of invariants for q = 2
in the natural representation of sl(2). In 1958, Burrow extended Wever’s formula to q > 1 and d = mq where m > 1.
In the present thesis, we concentrate on ﬁnding invariant Lie polynomials (simply called Lie invariants) in the natural representations of sl(2) and sl(3), and in the adjoint representation of sl(2). We ﬁrst review the method to construct the Hall basis of the free Lie algebra and the way to transform arbitrary Lie words into linear combinations of Hall words.
To ﬁnd the Lie invariants, we need to ﬁnd the nullspace of an integer matrix, and for this we use the Hermite normal form. After that, we review the generalized Witt dimension formula which can be used to compute the number of primitive Lie invariants of a given degree.
Secondly, we recall the result of Bremner on Lie invariants of degree ≤ 10 in the natural representation of sl(2). We extend these results to compute the Lie invariants of degree 12 and 14. This is the ﬁrst original contribution in the present thesis.
Thirdly, we compute the Lie invariants in the adjoint representation of sl(2) up to degree 8. This is the second original contribution in the present thesis.
Fourthly, we consider the natural representation of sl(3). This is a 3-dimensional natural representation of an 8-dimensional Lie algebra. Due to the huge number of Hall words in each degree and the limitation of computer hardware, we compute the Lie invariants only up to degree 12.
Finally, we discuss possible directions for extending the results. Because there
are inﬁnitely many diﬀerent simple ﬁnite dimensional Lie algebras and each of them
has inﬁnitely many distinct irreducible representations, it is an open-ended problem.
Advisors/Committee Members: Murray, Bremner, Chris, Soteros, Martin, John.

► A distance on a set is a comparative function. The smaller the distance between two elements of that set, the closer, or more similar, those…
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▼ A distance on a set is a comparative function. The smaller the distance between two elements of that set, the closer, or more similar, those elements are. Fr\'echet axiomatized the notion of distance into what is today known as a metric. In this thesis we study several generalizations of Fr\'echet's axioms. These include partial metric, strong partial metric, partial n-\mathfrak{M}etric and strong partial n-\mathfrak{M}etric. Those generalizations allow for negative distances, non-zero distances between a point and itself and even the comparison of n-tuples. We then present the scoring of a DNA sequence, a comparative function that is not a metric but can be modeled as a strong partial metric.
\ Using the generalized metrics mentioned above we create topological spaces and investigate convergence, limits and continuity in them. As an application, we discuss contractiveness in the language of our generalized metrics and present Banach-like fixed, common fixed and coincidence point theorems.
Advisors/Committee Members: Tymchatyn, Edward, Srinivasan, Raj, Szmigielski , Jacek, Martin, John, Dutchyn, Christopher.

► Traditional multiple hypothesis testing procedures, such as that of Benjamini and Hochberg, fix an error rate and determine the corresponding rejection region. In 2002 Storey…
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▼ Traditional multiple hypothesis testing procedures, such as that of Benjamini and Hochberg, fix an error rate and determine the corresponding rejection region. In 2002 Storey proposed a fixed rejection region procedure and showed numerically that it can gain more power than the fixed error rate procedure of Benjamini and Hochberg while controlling the same false discovery rate (FDR). In this thesis it is proved that when the number of alternatives is small compared to the total number of hypotheses, Storey’s method can be less powerful than that of Benjamini and Hochberg. Moreover, the two procedures are compared by setting them to produce the same FDR. The difference in power between Storey’s procedure and that of Benjamini and Hochberg is near zero when the distance between the null and alternative distributions is large, but Benjamini and Hochberg’s procedure becomes more powerful as the distance decreases. It is shown that modifying the Benjamini and Hochberg procedure to incorporate an estimate of the proportion of true null hypotheses as proposed by Black gives a procedure with superior power.
Multiple hypothesis testing can also be applied to regression diagnostics. In this thesis, a Bayesian method is proposed to test multiple hypotheses, of which the i-th null and alternative hypotheses are that the i-th observation is not an outlier versus it is, for i=1,...,m. In the proposed Bayesian model, it is assumed that outliers have a mean shift, where the proportion of outliers and the mean shift respectively follow a Beta prior distribution and a normal prior distribution. It is proved in the thesis that for the proposed model, when there exists more than one outlier, the marginal distributions of the deletion residual of the i-th observation under both null and alternative hypotheses are doubly noncentral t distributions. The “outlyingness” of the i-th observation is measured by the marginal posterior probability that the i-th observation is an outlier given its deletion residual. An importance sampling method is proposed to calculate this probability. This method requires the computation of the density of the doubly noncentral F distribution and this is approximated using Patnaik’s approximation. An algorithm is proposed in this thesis to examine the accuracy of Patnaik’s approximation. The comparison of this algorithm’s output with Patnaik’s approximation shows that the latter can save massive computation time without losing much accuracy.
The proposed Bayesian multiple outlier identification procedure is applied to some simulated data sets. Various simulation and prior parameters are used to study the sensitivity of the posteriors to the priors. The area under the ROC curves (AUC) is calculated for each combination of parameters. A factorial design analysis on AUC is carried out by choosing various simulation and prior parameters as factors. The resulting AUC values are high for various selected parameters, indicating that the proposed method can identify the majority of outliers within tolerable errors.…
Advisors/Committee Members: Mik, Bickis, Chris, Soteros, Murdoch, Duncan, Martin, John, Kusalik, Tony, Laverty, Bill, Srinivasan, Raj.

► We study here a model for a strand passage in a ring polymer about a randomly chosen location at which two strands of the polymer…
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▼ We study here a model for a strand passage in a ring polymer about a randomly chosen location at which two strands of the polymer have been brought gclose h together. The model is based on ƒ¦-SAPs, which are unknotted self-avoiding polygons in Z3 that contain a fixed structure ƒ¦ that forces two segments of the polygon to be close together. To study this model, the Composite Markov Chain Monte Carlo (CMCMC) algorithm, referred to as the CMC ƒ¦-BFACF algorithm, that I developed and proved to be ergodic for unknotted ƒ¦-SAPs in my M. Sc. Thesis, is used. Ten simulations (each consisting of 9.6 ~1010 time steps) of the CMC ƒ¦-BFACF algorithm are performed and the results from a statistical analysis of the simulated data are presented. To this end, a new maximum likelihood method, based on previous work of Berretti and Sokal, is developed for obtaining maximum likelihood estimates of the growth constants and critical exponents associated respectively with the numbers of unknotted (2n)-edge ƒ¦-SAPs, unknotted (2n)-edge successful-strand-passage ƒ¦-SAPs, unknotted (2n)-edge failed-strand-passage ƒ¦-SAPs, and (2n)-edge after-strand-passage-knot-type-K unknotted successful-strand-passage ƒ¦-SAPs. The maximum likelihood estimates are consistent with the result (proved here) that the growth constants are all equal, and provide evidence that the associated critical exponents are all equal.
We then investigate the question gGiven that a successful local strand passage occurs at a random location in a (2n)-edge knot-type K ƒ¦-SAP, with what probability will the ƒ¦-SAP have knot-type K f after the strand passage? h. To this end, the CMCMC data is used to obtain estimates for the probability of knotting given a (2n)-edge successful-strand-passage ƒ¦-SAP and the probability of an after-strand-passage polygon having knot-type K given a (2n)-edge successful-strand-passage ƒ¦-SAP. The computed estimates numerically support the unproven conjecture that these probabilities, in the n ¨ ‡ limit, go to a value lying strictly between 0 and 1. We further prove here that the rate of approach to each of these limits (should the limits exist) is less than exponential.
We conclude with a study of whether or not there is a difference in the gsize h of an unknotted successful-strand-passage ƒ¦-SAP whose after-strand-passage knot-type is K when compared to the gsize h of a ƒ¦-SAP whose knot-type does not change after strand passage. The two measures of gsize h used are the expected lengths of, and the expected mean-square radius of gyration of, subsets of ƒ¦-SAPs. How these two measures of gsize h behave as a function of a polygon fs length and its after-strand-passage knot-type is investigated.
Advisors/Committee Members: Soteros, Chris, Millett, K., Martin, John R., Laverty, William H., Bunt, Rick B., Srinivasan, Raj.

Szafron, M. L. (2009). Knotting statistics after a local strand passage in unknotted self-avoiding polygons in Z3. (Thesis). University of Saskatchewan. Retrieved from http://hdl.handle.net/10388/etd-04092009-231002

Note: this citation may be lacking information needed for this citation format:Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Szafron, Michael Lorne. “Knotting statistics after a local strand passage in unknotted self-avoiding polygons in Z3.” 2009. Thesis, University of Saskatchewan. Accessed March 21, 2019.
http://hdl.handle.net/10388/etd-04092009-231002.

Note: this citation may be lacking information needed for this citation format:Not specified: Masters Thesis or Doctoral Dissertation

Note: this citation may be lacking information needed for this citation format:Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Szafron ML. Knotting statistics after a local strand passage in unknotted self-avoiding polygons in Z3. [Thesis]. University of Saskatchewan; 2009. Available from: http://hdl.handle.net/10388/etd-04092009-231002

Note: this citation may be lacking information needed for this citation format:Not specified: Masters Thesis or Doctoral Dissertation

► Objective: Patient-reported outcomes (PROs) are measures collected from a patient to determine how he/she feels or functions in regards to a health condition. Longitudinal PROs,…
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▼ Objective: Patient-reported outcomes (PROs) are measures collected from a patient to determine how he/she feels or functions in regards to a health condition. Longitudinal PROs, which are collected at multiple occasions from the same individual, may be affected by response shift (RS). RS is a change in a person’s self-evaluation of a target construct. Latent variable models (LVMs) are statistical models that relate observed variables to latent variables (LV). LVMs are used to analyze PROs and detect RS. LVs are random variables whose realizations are not observable. Factor scores are estimates of LVs for each individual and can be estimated from parameter estimates of LVMs. Factor scoring methods to estimate factor scores include: Thurstone, Bartlett, and sum scores. This simulation study examines the effects of RS on factor scores used to test for change in the LV means and recommend a factor scoring method least affected by RS.
Methods: Data from two time points were fit to three confirmatory factor analysis (CFA) models. CFA models are a type of LVM. Each CFA model had different sets of parameters that were invariant over time. The unconstrained (Uncon) CFA model had no invariant parameters, the constrained (Con) model had all the parameters invariant, and the partially constrained (Pcon) model had some of the parameters invariant over time. Factor scores were estimated and tested for change over time via paired t-test. The Type I error, power, and factor loading (the regression coefficient between an observed and LV) and factor score bias were estimated to determine if RS influenced the test of change over time and factor score estimation.
Results: The results depended on the true LV mean. The Type I error and power were similar for all factor scoring methods and CFA models when the LV mean was 0 at time 1. For LV mean of 0.5 at time 1 the Type I error and power increased as RS increased for all factor scores except for scores estimated from the Uncon model and Bartlett method. The biases of the factor loadings were unaffected by RS when estimated from an Uncon model. The factor scores estimated from the Uncon model and the Bartlett and sum scores method had the smallest factor score biases.
Conclusion: The factor scores estimated from the Uncon model and the Bartlett method was least affected by RS and performed best in all measures of Type I error, statistical power, factor loading and factor score bias. Estimating factor scores from PROs data that ignores RS may result in erroneous (or biased) estimates.
Advisors/Committee Members: Lix, Lisa, Liu, Juxin, Li, Longhai, Sarty, Gordon, Martin, John.

► With the rapid development of cellular communication techniques, many recent studies have focused on improving the quality of service (QoS) in cellular networks. One characteristic…
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▼ With the rapid development of cellular communication techniques, many recent studies have focused on improving the quality of service (QoS) in cellular networks. One characteristic of the systems in cellular networks, which can have direct impact on the system QoS, is the fluctuation of the system capacity. In this thesis, the QoS of systems with capacity fluctuations is studied from two perspectives: (1) priority queueing systems with preemption, and (2) the M/M/~C/~C system.
In the first part, we propose two models with controlled preemption and analyze their performance in the context of a single reference cell that supports two kinds of traffic (new calls and handoff calls). The formulae for calculating the performance measures of interest (i.e., handoff call blocking probability, new call blocking and dropping probabilities) are developed, and the procedures for solving optimization problems for the optimal number of channels required for each proposed model are established. The proposed controlled preemption models are then compared to existing non-preemption and full preemption models from the following three perspectives: (i) channel utilization, (ii) low priority call (i.e., new calls) performance, and (iii) flexibility to meet various constraints. The results showed that the proposed controlled preemption models are the best models overall.
In the second part, the loss system with stochastic capacity, denoted by M/M/~C/~C, is analyzed using the Markov regenerative process (MRGP) method. Three different distributions of capacity interchange times (exponential, gamma, and Pareto) and three different capacity variation patterns (skip-free, distance-based, and uniform-based) are considered. Analytic expressions are derived to calculate call blocking and dropping probabilities and are verified by call level simulations. Finally, numerical examples are provided to determine the impact of different distributions of capacity interchange times and different capacity variation patterns on system performance.
Advisors/Committee Members: Srinivasan, Raj, Martin, John, Bickis, Mik, Soteros, Chris, Sparks, Gordon A., Hlynka, Myron.

► Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented…
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▼ Positivity of polynomials, as a key notion in
real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set KS is compact, then any polynomial which is strictly positive on KS is contained in the preordering TS.Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module MS is archimedean, any polynomial which is strictly positive on KS is contained in MS, i.e., has a linear representation.In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: KS compact and MS archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.Secondly, we investigate the relationship of MS and TS. We use elementary arguments to prove in the one variable case when KS is compact, they are equal.Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on KS to be contained in MS in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.
Advisors/Committee Members: Marshall, Murray, Kuhlmann, Salma, McQuillan, Ian, Martin, John R., Bremner, Murray R..

► The dynamics of a single underwater vehicle in an ideal irrotational fluid may be modeled by a Lagrangian system with configuration space the Euclidean group.…
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▼ The dynamics of a single underwater vehicle in an ideal irrotational fluid may be modeled by a Lagrangian system with configuration space the Euclidean group. If hydrodynamic coupling is ignored then two coupled vehicles may be modeled by the direct product of two single-vehicle systems. We consider this system in the case that the vehicles are coupled mechanically, with an ideal spherically symmetric joint, finding all of the relative equilibria. We demonstrate that there are relative equilibria in certain novel momentum-generator regimes identified by Patrick et.al. "Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods", Arch. Rational Mech. Anal., 174:301 – 344, 2004.
Advisors/Committee Members: Patrick, George W., Soteros, Chris, Rangacharyulu, Chilakamarri (Chary), Martin, John R., Szmigielski, Jacek.

► This thesis is about multiple hypothesis testing and its relation to the P-value. In Chapter 1, the methodologies of hypothesis testing among the three inference…
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▼ This thesis is about multiple hypothesis testing and its relation to the P-value. In Chapter 1, the methodologies of hypothesis testing among the three inference schools are reviewed. Jeffreys, Fisher, and Neyman advocated three different approaches for testing by using the posterior probabilities, P-value, and Type I error and Type II error probabilities respectively. In Berger's words ``Each was quite critical of the other approaches." Berger proposed a potential methodological unified conditional frequentist approach for testing. His idea is to follow Fisher in using the P-value to define the strength of evidence in data and to follow Fisher's method of conditioning on strength of evidence; then follow Neyman by computing Type I and Type II error probabilities conditioning on strength of evidence in the data, which equal the objective posterior probabilities of the hypothesis advocated by Jeffreys. Bickis proposed another estimate on calibrating the null and alternative components of the distribution by modeling the set of P-values as a sample from a mixed population composed of a uniform distribution for the null cases and an unknown distribution for the alternatives. For tackling multiplicity, exploiting the empirical distribution of P-values is applied. A variety of density estimators for calibrating posterior probabilities of the null hypothesis given P-values are implemented. Finally, a noninterpolatory and shape-preserving estimator based on B-splines as smoothing functions is proposed and implemented.
Advisors/Committee Members: Bickis, Mikelis G., Srinivasan, Raj, Soteros, Chris, Martin, John R., Kelly, Ivan W..

► There is a new approach in dimension theory, proposed by A. N. Dranish­nikov and based on the concept of extension types of complexes. Following Dranishnikov,…
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▼ There is a new approach in dimension theory, proposed by A. N. Dranish­nikov and based on the concept of extension types of complexes. Following Dranishnikov, for a CW-complex L we introduce the definition of exten­sion type [L] of this complex. Further, for a space X we define the notion of extension dimension e - dim of X, which generalizes both Lebesgue and cohomological dimensions.
An adequate homotopy and shape theories, which are specifically designed to work for at most [L]-dimensional spaces, have also been developed. Fol­lowing A. Chigogidze, we present the concept of [L]-homotopy. This concept generalizes the concept of standard homotopy as well as of n-homotopy, intro­duced by R. H. Fox and studied by J.H.C. Whitehead. We also investigate the class of spaces which play a significant role in [L]-homotopy theory, namely, absolute (neighborhood) extensors modulo a complex (shortly A(N)E([L])-­spaces). Observe that A(N)E([Sn])-spaces are precisely A(N)E(n)-spaces. The first result of the present thesis describes A(N)E([L])-spaces in terms of local properties and provides an extension-dimensional version of Dugundji theorem.
Another result of the present work is related to the theory of continuous selections. The finite-dimensional selection theorem of E. Michael is very useful in geometric topology and is one of the central theorems in the theory of continuous selections of multivalued mappings. In the thesis we present the proof of an extension-dimensional version of the finite dimensional selection theorem. This version contains Michael's original finite dimensional theorem as a special case.
The concept of [L]-homotopy naturally leads us to the definition of alge­braic [L]-homotopy invariants, and, in particular, [L]-homotopy groups. We give a detailed description of [L]-homotopy groups introduced by Chigogidze.
The notion of closed model category, introduced by D. Quillen, gives an axiomatic approach to homotopy theory. It should be noted that while there exist several important examples of closed model category structures on the category of topological spaces TOP, the associated homotopies in all cases are very closely related to the ordinary homotopy. Based on the above men­tioned [L]-homotopy groups we, in this thesis, provide the first examples of model category structures on TOP whose homotopies are substantially dif­ferent from the ordinary one. Namely, we show that [L]-homotopy is indeed a homotopy in the sense of Quillen for each finite CW-complex L.
Observe that [L]-homotopy groups may differ from the usual homotopy groups even for polyhedra. The problem which arises in a natural way is to describe [L]-homotopy groups in terms of "usual" algebraic invariants of X and L (in particular, in terms of homotopy and homology groups). In the present work we compute the n-th [L]-homotopy group of Sn for a complex L whose extension type lies between extension types of Sn and Sn+l.
Advisors/Committee Members: Chigogidze, Alexander, Marshall, Murray, Koustov, Alexandre V. (Sasha), Khoshkam, Mahmood, Dranishnikov, Alexander, Martin, John R., Tymchatyn, Edward D..